Edited by
C.M. Dafermos, Brown University, Providence, RI, USA
Eduard Feireisl, Mathematical Institute AS CR, Prague, Czech Republic.
Description
The material collected in this volume reflects the active present of this area of mathematics, ranging from the abstract theory of gradient
flows to stochastic representations of non-linear parabolic PDE's.
Articles will highlight the present as well as expected future directions
of development of the field with particular emphasis on applications.
The article by Ambrosio and Savaré discusses
the most recent
development in the theory of gradient flow of probability measures. After an introduction reviewing the properties of the Wasserstein
space and corresponding subdifferential calculus, applications are given to evolutionary
partial differential equations. The contribution
of Herrero provides a description of some mathematical approaches developed to account for quantitative as well as qualitative aspects
of chemotaxis. Particular attention is paid to the limits of cell's
capability to measure external cues on the one hand, and to provide
an overall description of aggregation models for the slim mold
Dictyostelium discoideum on the other.
The chapter written
by Masmoudi deals with a rather different topic - examples of singular limits in hydrodynamics. This is nowadays a well-studied issue
given the amount of new results based on the development of the existence theory for rather general systems of equations in hydrodynamics.
The paper by DeLellis addreses the most recent results for the transport equations with regard to possible applications in the theory
of hyperbolic systems of conservation laws. Emphasis is put on the development of the theory in the case when the governing field is
only a BV function.
The chapter by Rein represents a comprehensive survey of results on the Poisson-Vlasov system in astrophysics. The
question of global stability of steady states is addressed in detail. The contribution of Soner is devoted to different representations
of non-linear parabolic equations in terms of Markov processes. After a brief introduction on the linear theory, a class of
non-linear
equations is investigated, with applications to stochastic control and differential games.
The chapter written by Zuazua presents some
of the recent progresses done on the problem of controllabilty of partial differential equations. The applications include the linear
wave and heat equations,parabolic equations with coefficients of low regularity, and some fluid-structure interaction models.
Audience:
University libraries and Research mathematicians
